Galois extension

In mathematics, a Galois extension is an algebraic field extension E/F that is normal and separable; or equivalently, E/F is algebraic, and the field fixed by the automorphism group Aut(E/F) is precisely the base field F. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory. 

A result of Emil Artin allows one to construct Galois extensions as follows: If E is a given field, and G is a finite group of automorphisms of E with fixed field F, then E/F is a Galois extension.

Characterization of Galois extensions

An important theorem of Emil Artin states that for a finite extension E/F, each of the following statements is equivalent to the statement that E/F is Galois:

Other equivalent statements are:

• Every irreducible polynomial in F[x] with at least one root in E splits over E and is separable.
• |Aut(E/F)| ≥ [E:F], that is, the number of automorphisms is at least the degree of the extension.
• F is the fixed field of a subgroup of Aut(E).
• F is the fixed field of Aut(E/F).
• There is a one-to-one correspondence between subfields of E/F and subgroups of Aut(E/F).

Examples

There are two basic ways to construct examples of Galois extensions.

• Take any field E, any subgroup of Aut(E), and let F be the fixed field.
• Take any field F, any separable polynomial in F[x], and let E be its splitting field.

Adjoining to the rational number field the square root of 2 gives a Galois extension, while adjoining the cube root of 2 gives a non-Galois extension. Both these extensions are separable, because they have characteristic zero. The first of them is the splitting field of x2 − 2; the second has normal closure that includes the complex cube roots of unity, and so is not a splitting field. In fact, it has no automorphism other than the identity, because it is contained in the real numbers and x3 − 2 has just one real root. For more detailed examples, see the page on the fundamental theorem of Galois theory

An algebraic closure of an arbitrary field is Galois over if and only if is a perfect field.

References

1. See the article Galois group for definitions of some of these terms and some examples.